Mean perimeter of the convex hull of a random walk in a semi-infinite medium

We study various properties of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory, in the presence of an infinite reflecting wall. Recently [Phys. Rev. E 91, 050104(R) (2015)], we announced that the mean perimeter of the convex hull at time $t$, rescaled by $\sqrt{Dt}$, is a nonmonotonous function of the initial distance to the wall. In this article, we first give all the details of the derivation of this mean rescaled perimeter, in particular its value when starting from the wall and near the wall. We then determine the physical mechanism underlying this surprising nonmonotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull. Finally, we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the Brownian motion as a function of the initial distance to the wall.

Figure 1

Minimal model of a Brownian motion in the presence of a reflecting infinite wall, that starts at a distance $d$ from the wall. The trajectory is represented by the thin red path, and the convex hull of the Brownian motion, defined as the minimum convex polygon enclosing the trajectory, is the thick green polygon.

Figure 2

Definition of the maximum ${\mathcal{M}}^{\left(d\right)}(\theta ,t)$ of the trajectory in the direction $\theta$ at time $t$ for a Brownian motion starting at a distance $d$ from the reflecting wall. It can be obtained by introducing a plane orthogonal to the direction $\theta$ far away from the trajectory, moving it along the direction $\theta$ towards the trajectory until touching it, and measuring the distance between the starting point and this plane.

Figure 3

Definition of the geometry in a wedge with one reflecting edge and one absorbing edge.

Figure 4

Equivalence between a wedge with an absorbing edge and a reflecting edge and a wedge of double top angle with two absorbing edges. The starting point is at the polar coordinates $(r_0,{\phi }_0)$.

Figure 5

Plot of the exact expression of the mean rescaled maximum and of the corresponding development given in Eq. (49), for $\theta =-\pi /5$ (left) and $\theta =-4\pi /9$ (right). The range of validity of the development becomes smaller when $\theta$ approaches $-\pi /2$.

Figure 6

Plot of the mean rescaled perimeter of the convex hull of a Brownian motion in the presence of a reflecting wall as a function of the rescaled initial distance $x$, compared to numerical simulations. The main feature of this observable is its minimum with respect to the rescaled initial distance to the wall. Some details on the numerical evaluation and simulations are given in Secs. 2d and 2e.

Figure 7

Schematic representation of the convex hull, divided into two parts, an inward part in red and an outward part in light green. The wall has two effects: an effect of reduction of the accessible space that acts on the inward part of the convex hull, and an effect of effective repulsion that impacts on the outward part of the convex hull. (a) $d\gg \sqrt{Dt}$: the trajectories feel no effect of the wall, the two parts of the convex hull have the same length. (b) $d\sim \sqrt{Dt}$: the effect of reduction of the accessible cuts the inward part of the convex hull. (c) $d\ll \sqrt{Dt}$: the length of the inward part is still lower, but the effect of effective repulsion makes the outward part more rounded, increasing its length.

Figure 8

Rescaled length of the inward (in red) and outward (in light green) parts of the convex hull with respect to the rescaled initial distance to the wall, obtained from Eqs. (63) and (64). The black horizontal line represents the length of these two parts in the absence of confinement ($2\sqrt{\pi }$).

Figure 9

Plot of the mean rescaled maximum in the direction $\theta$ parametrized by the angle $\theta$, obtained from Eqs. (5), (61), and (62). The dots represent the mean rescaled maximum for three values of the initial rescaled distance, and the plain circle gives the mean rescaled maximum for a nonconfined walk (which is $2/\sqrt{\pi }\simeq 1.13$).

Figure 10

Definition of the extension of the convex hull on the wall $\mathcal{E}(d,t)$ at time $t$ for a Brownian motion starting at a distance $d$ from the wall, represented by the plain blue segment along the wall.

Figure 11

(a) The probability that the downward semiextension ${\mathcal{E}}_{\mathrm{half}}$ is smaller than $m$ is the survival probability in the presence of the semi-infinite absorbing thick blue line. (b) This probability is exactly the survival probability in an infinite absorbing wedge of top angle $2\pi$ for a starting point located at a distance $m$ from the apex of the wedge with an angle ${\phi }_0=\pi$.

Figure 12

Plot of the analytical expression (74) of the mean extension at time $t$ of a planar Brownian motion on the reflecting wall rescaled by $\sqrt{Dt}$ as a function of the rescaled initial distance $x=d/\sqrt{Dt}$.

Figure 13

Brownian motion in an infinite absorbing planar wedge of top angle $\alpha$, described in polar coordinates and starting from the point $(r_0,{\phi }_0)$.

Figure 14

Definition of the points, distances, and angles used in the derivation of the geometric relations between $(r_0,{\phi }_0)$ and $(M,\theta )$, in the case $\theta >0$ (a) and $\theta <0$ (b). $A$ is the starting point.